Q.E.D.

34 35 Power sums proofs by double counting The marvellous Pythagorean oneglance dissection proof below shows that the sum of the first n natural numbers is half the number of pebbles in the rectangle, that is, nn 12. Carl Friedrich Gauss 1777 1855, one of the giants of mathematics, rediscovered this formula at the age of ten. Asked by his teacher to sum up the first 100 natural numbers, he made short work of the tedious task by observing that 1 100 2 99 ... 50 51 101, and that therefore the required sum was 50 . 101 5050. This reasoning corresponds to looking carefully at the rectangle shown below, row by row 1 4 2 3 5 and the sum is 2 . 5 10. The first diagram on the right elegantly shows that three times the sum of the first n squares equals the number of pebbles in the rectangle, that is, 2n 11 2 ... n. The second diagram on the right demonstrates that the sum of the first n cubes is equal to the sum of the first n natural numbers squared. For formul for the sums of the first n fourth powers, fifth powers, and so on, see page 55.